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shscouplingmatrix.doc
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shscouplingmatrix.doc
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This routine returns the spherical harmonic coupling matrix for a given set of
Slepian basis functions. This matrix relates the power spectrum expectation of
the function expressed in a subset of the best-localized Slepian functions to
the expectation of the global power spectrum.
Usage
-----
kij = SHSCouplingMatrix (galpha, nmax)
Returns
-------
kij : float, dimension (lmax+1, lmax+1)
The coupling matrix that relates the power spectrum expectation of the
function expressed in a subset of the best-localized Slepian functions to
the expectation of the global power spectrum.
Parameters
----------
galpha : float, dimension ((lmax+1)**2, nmax)
An array of Slepian functions, arranged in columns from best to worst
localized.
nmax : input, integer
The number of Slepian functions used in reconstructing the function.
Description
-----------
SHSCouplingMatrix returns the spherical harmonic coupling matrix that relates
the power spectrum expectation of the function expressed in a subset of the
best-localized Slepian functions to the expectation of the global power spectrum
(assumed to be stationary). The Slepian functions are determined by a call to
either (1) SHReturnTapers and then SHRotateTapers, or (2) SHReturnTapersMap.
Each row of galpha contains the (lmax+1)**2 spherical harmonic coefficients of a
Slepian function that can be unpacked using SHVectorToCilm. The Slepian
functions must be normalized to have unit power (that is the sum of the
coefficients squared is 1).
The relationship between the global and localized power spectra is given by:
< S_{\tilde{f}}(l) > = \sum_{l'=0}^lmax K_{ll'} S_{f}(l')
where S_{\tilde{f}} is the expectation of the power spectrum at degree l of the
function expressed in Slepian functions, S_{f}(l') is the expectation of the
global power spectrum, and < ... > is the expectation operator. The coupling
matrix is given explicitly by
K_{ll'} = \frac{1}{2l'+1} Sum_{m=-l}^l Sum_{m'=-l'}^l' ( Sum_{alpha=1}^nmax
g_{l'm'}(alpha) g_{lm}(alpha) )**2